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 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 5 Convex Analysis.
 A. Basic concepts of convex analysis.
 B. Caratheodory's theorem.
 C. Relative interior.
 D. Recession cone.
 E. Intersection of nested convex sets.
 F. Preservation of closeness under linear transformation.
 G. Weierstrass Theorem.
 H. Local minima of convex function.
 I. Projection on convex set.
 J. Existence of solution of convex optimization problem.
 K. Partial minimization of convex functions.
 L. Hyperplanes and separation.
 M. Nonvertical separation.
 N. Minimal common and maximal crossing points.
 O. Minimax theory.
 P. Saddle point theory.
 Q. Polar cones.
 R. Polyhedral cones.
 S. Extreme points.
 T. Directional derivative and subdifferential.
 U. Feasible direction cone, tangent cone and normal cone.
 V. Optimality conditions.
 W. Lagrange multipliers for equality constraints.
 X. Fritz John optimality conditions.
 Y. Pseudonormality.
 Z. Lagrangian duality.
 a. Geometric multipliers.
 b. Dual problem.
 c. Connection of dual problem with minimax theory.
 [. Conjugate duality.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Connection of dual problem with minimax theory.

roposition

1. The problem ( Primal problem ) is equivalent to

2. The problem ( Dual problem ) is equivalent to

Proof

Note that The rest follows from the definitions of the problems ( Primal problem ) and ( Dual problem ).

Proposition

(Necessary and sufficient optimality conditions).The vectors form a solution of the problem ( Primal problem ) and a geometric multiplier pair if and only if the following four conditions hold.

 (Primal feasibility)
 (Dual feasibility)
 (Lagrangian optimality)
 (Complementary slackness)

Proof

If the form a solution of the problem ( Primal problem ) and a geometric multiplier pair then the statements ( Primal feasibility ) and ( Dual feasibility ) follow from the definitions and ( Lagrangian optimality ),( Complementary slackness ) follow from the proposition ( Geometric multiplier property ).

Conversely, using the conditions of the theorem we obtain The the equality follows from the propositions ( Visualization lemma ) and ( Weak duality theorem ).

 Notation. Index. Contents.
 Copyright 2007